A rectangular beam is to be cut from the log into a circular cross section. If the strength of the beam is proportional to the width and the square of the depth, find the dimensions that will give the strongest beam
I assume we can center the beam in the circle. So, let
2x = width
2y = depth
That means that x^2+y^2 = r^2, the radius of the log
If we let the strength be z, then we know that
z = kxy^2 = kx(r^2-x^2) = kr^2x - kx^3
dz/dx = k(r^2 - 3x^2)
Now just find x where dz/dx=0
You can see that z will be a maximum.
To find the dimensions that will give the strongest beam, we need to maximize the strength of the beam, which is proportional to the width and the square of the depth.
Let's assume the rectangular beam has a width (W) and a depth (D). We want to convert this rectangular beam into a circular cross section while maximizing its strength.
The area of a rectangular beam is given by A = W * D, and the strength is proportional to the width (W) and the square of the depth (D^2), so we can express the strength (S) as S = k * W * D^2, where k is a constant of proportionality.
Now, we need to relate the dimensions (W, D) of the rectangular beam to the diameter (Diameter) of the circular cross section.
For a circular cross section, the area (A') is given by A' = π * (Diameter/2)^2 = π * (D/2)^2, where D is the diameter of the circular cross section.
To maximize the strength of the beam, we want to maximize the value of S = k * W * D^2.
To simplify the problem, we can write the width (W) of the rectangular beam in terms of the diameter (D) of the circular cross section. Since the width is assumed to be equal to the diameter (W = D), we can substitute W with D in the expression for strength:
S = k * W * D^2 = k * D * D^2 = k * D^3
Now, we have a single variable, the diameter (D), to represent the dimensions of the beam.
To find the maximum strength, we can take the derivative of S with respect to D and set it equal to zero:
dS/dD = 3k * D^2 = 0
Solving for D, we find D = 0.
However, a diameter of 0 does not make sense physically, so our assumption that the width equals the diameter must be incorrect.
Therefore, the rectangular beam cannot be cut into a circular cross section to achieve the strongest beam. We need to reconsider the shape or dimensions of the beam to maximize its strength.